📚 IGCSE Geometry 5: Circle Theorems and Trigonometry | IGCSE 几何 5：圆定理与三角学
This revision guide covers the essential circle theorems and trigonometry skills required for IGCSE Mathematics. You will learn how to identify angle relationships in circles, apply tangent properties, and solve problems using right-angled and non-right-angled triangle methods. The content is suitable for both Core and Extended candidates, with Extended-only tools such as the sine rule, cosine rule, and 3D trigonometry clearly marked where they appear.
本复习指南涵盖 IGCSE 数学中必需的圆定理与三角学技能。你将学习如何识别圆中的角关系、应用切线性质，并使用直角三角形和非直角三角形方法解决问题。内容同时适用于核心课程和扩展课程，扩展课程专属工具如正弦定理、余弦定理和三维三角学会在相关位置明确标注。
1. Angle Facts and Parallel Lines | 角的基本性质与平行线
Before working with circles and trigonometry, you must be confident with basic angle facts. Angles on a straight line add up to 180°, angles around a point add up to 360°, and vertically opposite angles are equal.
在学习圆和三角函数之前，你必须熟练掌握基本的角性质。直线上的角之和为 180°，一点周围的角之和为 360°，对顶角相等。
When two parallel lines are cut by a transversal, corresponding angles are equal, alternate angles are equal, and interior angles on the same side add up to 180°. These facts often appear as hidden steps in harder circle-theorem proofs.
当两条平行线被一条截线所截时，同位角相等，内错角相等，同旁内角之和为 180°。这些性质往往作为隐藏步骤出现在较难的圆定理证明中。
- Corresponding angles are in matching positions. | 同位角处于对应位置。
- Alternate angles are inside the parallel lines but on opposite sides of the transversal. | 内错角位于两条平行线之间，但在截线的两侧。
- Co-interior angles are inside the parallel lines on the same side of the transversal and sum to 180°. | 同旁内角位于两条平行线之间且在同一侧，和为 180°。
Always mark these angles on the diagram first. This helps you see hidden connections before applying circle theorems.
始终先在图上标出这些角。这有助于你在应用圆定理之前发现隐藏的联系。
2. Triangles and Pythagoras' Theorem | 三角形与勾股定理
A triangle's interior angles always sum to 180°. An equilateral triangle has three 60° angles, an isosceles triangle has two equal sides and two equal base angles, and a right-angled triangle contains one 90° angle.
三角形的内角和始终为 180°。等边三角形有三个 60° 角，等腰三角形有两条相等的边和两个相等的底角，直角三角形包含一个 90° 角。
Pythagoras' theorem states that in any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This is written as:
勾股定理指出，在任何直角三角形中，斜边的平方等于另外两条边的平方之和。它写作：
c² = a² + b²
Here c is the hypotenuse, the side opposite the right angle. You can use the rule to find a missing side, and its converse helps you test whether a triangle is right-angled.
这里的 c 是斜边，也就是直角所对的边。你可以用这个公式求缺失的边，它的逆定理可以帮助你检验一个三角形是否为直角三角形。
o shorter sides are 6 cm and 8 cm, then c² = 6² + 8² = 36 + 64 = 100, so c = √100 = 10 cm.
例如，如果两条较短边为 6 cm 和 8 cm，那么 c² = 6² + 8² = 36 + 64 = 100，所以 c = √100 = 10 cm。
3. Circle Terminology | 圆的基本术语
You must know the exact meaning of the main parts of a circle. A radius is a line segment from the centre to any point on the circumference. A diameter is a chord that passes through the centre and has length twice the radius.
你必须了解圆各主要部分的准确含义。半径是从圆心到圆周上任意一点的线段。直径是经过圆心的一条弦，其长度是半径的两倍。
A chord is any line segment joining two points on the circumference. An arc is a portion of the circumference, and a sector is the region enclosed by two radii and an arc. A segment is the region enclosed by a chord and an arc.
弦是连接圆周上两点的任意线段。弧是圆周的一部分，扇形是由两条半径和一段弧围成的区域。弓形是由一条弦和一段弧围成的区域。
A tangent is a straight line that touches the circle at exactly one point. A secant, by contrast, cuts the circle at two points. Understanding these terms will help you read exam diagrams accurately.
切线是与圆恰好相切于一点的直线。相比之下，割线是在两点处截穿圆的直线。理解这些术语将帮助你准确阅读考试图形。
4. Circle Theorem 1: Angle at Centre | 圆定理 1：圆心角
The first circle theorem states that the angle subtended by an arc at the centre of the circle is twice the angle subtended by the same arc at any point on the circumference.
第一个圆定理指出，同一段弧所对的圆心角是同一段弧所对的圆周角的两倍。
∠AOB = 2 × ∠APB
Here O is the centre, A and B lie on the circumference, and P is any other point on the major arc. The angle at O is double the angle at P.
这里 O 为圆心，A 和 B 位于圆周上，P 是优弧上的任意另一点。O 处的角是 P 处角的两倍。
= 60°. This theorem is often used with the angle sum of a triangle to find missing angles.
如果圆心角为 120°，那么圆周角为 120° ÷ 2 = 60°。该定理常与三角形的内角和一起使用，以求出缺失的角。
- Check that both angles stand on the same arc. | 检查两个角是否位于同一段弧上。
- Make sure the vertex of the circumference angle actually lies on the circle. | 确保圆周角的顶点确实位于圆上。
5. Circle Theorem 2: Angle in a Semicircle | 圆定理 2：半圆中的圆周角
ates a 180° angle at the centre, so the angle at the circumference must be 180° ÷ 2 = 90°.
半圆中的圆周角始终是直角。这来自圆心角定理：直径在圆心处形成 180° 角，因此圆周上的角必定是 180° ÷ 2 = 90°。
In exams, the diameter is often not drawn as an obvious straight line. Look for the phrase 'AB is a diameter' or a mark showing that O lies on a chord. Then you know any angle drawn from the two ends of that diameter to the circumference is 90°.
在考试中，直径往往不会画成一条明显的直线。注意短语 'AB 是直径' 或表明 O 在弦上的标记。那么你就知道从直径两端到圆周上任意一点所成的角为 90°。
This theorem is very powerful in problems involving right-angled triangles inside circles, because it allows you to apply Pythagoras' theorem and basic trigonometry.
该定理在涉及圆内直角三角形的题目中非常有用，因为它允许你应用勾股定理和基本三角函数。
6. Circle Theorem 3: Cyclic Quadrilaterals | 圆定理 3：圆内接四边形
A cyclic quadrilateral is a four-sided shape with all four vertices lying on the circumference of a circle. For any cyclic quadrilateral, opposite angles add up to 180°.
圆内接四边形是四个顶点都位于圆周上的四边形。对于任何圆内接四边形，对角之和为 180°。
∠A + ∠C = 180° and ∠B + ∠D = 180°
se the exterior angle of a cyclic quadrilateral: an exterior angle is equal to the interior opposite angle.
如果一个角为 85°，那么对角必定是 180° - 85° = 95°。你也可以使用圆内接四边形的外角：外角等于与其相对的内角。
This is a classic exam theorem because the shape may be presented in a nontraditional orientation. Always check whether all four vertices lie on the circle before applying the rule.
这是考试中的经典定理，因为图形可能以非传统的方向呈现。在应用该规则之前，一定要先检查四个顶点是否都在圆上。
7. Tangent and Radius | 切线与半径
A tangent to a circle is perpendicular to the radius at the point of contact. This means that the angle between a tangent and the radius drawn to the point where the tangent touches the circle is exactly 90°.
圆的切线垂直于经过切点的半径。这意味着切线与经过切点的半径之间的角恰好是 90°。
∠OTP = 90°
Here O is the centre, T is the point of contact, and P is any point on the tangent. This right angle is the starting point for many problems involving Pythagoras and trigonometry.
这里 O 为圆心，T 为切点，P 为切线上的任意一点。这个直角是许多涉及勾股定理和三角学的题目的起点。
nts drawn to a circle are equal in length. If PA and PB are tangents from P, then PA = PB.
还有一个从一点引切线的定理：从圆外一点引圆的两条切线长度相等。如果 PA 和 PB 是从点 P 引出的切线，那么 PA = PB。
In more advanced problems, you may need to identify two right-angled triangles sharing a common side. Use this structure to build equations for missing lengths.
在更高级的题目中，你可能需要识别两个共享同一条边的直角三角形。利用这种结构建立关于缺失长度的方程。
8. Introduction to Trigonometry | 三角学入门
Trigonometry connects the angles of a right-angled triangle to the ratios of its sides. The three basic ratios are sine, cosine, and tangent, usually abbreviated as sin, cos, and tan.
三角学将直角三角形的角与其边之间的比联系起来。三个基本比分别是正弦、余弦和正切，通常缩写为 sin、cos 和 tan。
sin θ = opposite ÷ hypotenuse
cos θ = adjacent ÷ hypotenuse
tan θ = opposite ÷ adjacent
The mnemonic SOH CAH TOA helps you remember these definitions. The hypotenuse is always the longest side, opposite the right angle. The opposite side faces the angle θ, and the adjacent side is next to θ but is not the hypotenuse.
助记口诀 SOH CAH TOA 可以帮助你记住这些定义。斜边始终是最长边，位于直角对面。对边面对角 θ，邻边紧邻 θ 但不是斜边。
| θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
You should memorise these exact values for 30°, 45°, and 60°. They often appear in calculations involving surds and in circle-theorem contexts.
你应该记住 30°、45° 和 60° 的这些精确值。它们经常出现在涉及根式的计算和圆定理的背景中。
9. Sine and Cosine Rules | 正弦定理与余弦定理
For non-right-angled triangles, the sine rule and cosine rule are essential tools for Extended IGCSE candidates. The sine rule links side lengths to the sines of the opposite angles:
对于非直角三角形，正弦定理和余弦定理是 IGCSE 扩展课程考生的必备工具。正弦定理将边长与对角的正弦联系起来：
a/sin A = b/sin B = c/sin C
Here the side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. Use the sine rule when you know two angles and one side, or two sides and a non-included angle.
这里的边 a 对角 A，边 b 对角 B，边 c 对角 C。当你知道两个角和一个边，或者两个边和一个非夹角时，使用正弦定理。
The cosine rule connects all three sides with one angle. It is often written as:
余弦定理将三条边与一个角联系起来。它通常写作：
a² = b² + c² - 2bc cos A
Use the cosine rule when you know three sides and want to find an angle, or when you know two sides and the included angle. You can rearrange it to solve for angle A:
当你知道三条边并想求一个角，或者知道两条边及其夹角时，使用余弦定理。你可以将它变形以求解角 A：
cos A = (b² + c² - a²) ÷ (2bc)
le cases, check whether there may be two possible triangles because sin θ = sin(180° - θ).
在代入之前，一定要清楚地标记你的三角形。在正弦定理存在歧义的情况下，要检查是否可能有两个三角形，因为 sin θ = sin(180° - θ)。
10. Area of a Triangle and Applications | 三角形面积与应用
The area of any non-right-angled triangle can be found using the formula:
任何非直角三角形的面积都可以用以下公式求得：
Area = 1/2 × ab sin C
Here a and b are two sides, and C is the included angle. This formula is especially useful when the perpendicular height is not known directly.
这里的 a 和 b 是两条边，C 是它们的夹角。当垂直高度无法直接得知时，这个公式尤其有用。
In application problems, you may need to combine the sine rule, cosine rule, and area formula in several stages. Read the whole question, identify which triangle information is given, and choose the correct rule step by step.
在应用题中，你可能需要在几个阶段中结合使用正弦定理、余弦定理和面积公式。阅读整道题目，确定给出了哪些三角形信息，然后逐步选择正确的规则。
Bearings, navigation, and real-life measurement tasks are common IGCSE contexts. Often the diagram includes a circle or tangent line alongside triangle data, so circle theorems and trigonometry must be used together.
方位角、导航和实际测量任务是 IGCSE 的常见情境。图形通常会在三角形数据旁边包含圆或切线，因此圆定理和三角学必须结合使用。
11. 3D Trigonometry | 三维三角学
Extended IGCSE candidates may be asked to solve problems in three dimensions, such as finding an angle between a line and a plane, or the angle between two planes. Start by identifying a right-angled triangle that contains the required angle.
扩展课程考生可能需要解决三维问题，例如求直线与平面之间的角，或两个平面之间的角。首先确定一个包含所求角的直角三角形。
A common strategy is to work in two stages: first find a necessary length in one triangle using Pythagoras or trigonometry, then use that length in a second triangle to find the required angle.
一种常见策略是分两个阶段：首先使用勾股定理或三角函数在一个三角形中求出必要的边长，然后在第二个三角形中使用该边长求出所需的角。
Draw the extracting right triangle separately from the 3D diagram. This makes it much easier to label the opposite, adjacent, and hypotenuse, and to avoid choosing the wrong plane.
将提取出的直角三角形与三维图形分开绘制。这样更容易标出对边、邻边和斜边，并避免选错平面。
- Identify the line segment and the plane carefully. | 仔细辨别线段与平面。
- Use the projection of the line onto the plane to locate the right angle. | 利用直线在平面上的投影来确定直角。
- Keep intermediate answers in exact form when possible, such as √48, to maintain accuracy. | 在可能的情况下保留中间答案的根式形式，如 √48，以保持精度。
12. Common Mistakes and Exam Tips | 常见错误与考试提示
One common error is applying circle theorems without checking that the points actually lie on the same circle. Do not assume a quadrilateral is cyclic unless the question states it or its vertices all lie on the circumference.
一个常见错误是在未检查各点是否确实位于同一个圆上的情况下就应用圆定理。除非题目明确说明四边形的顶点都在圆周上，否则不要假定它是圆内接四边形。
In trigonometry, students sometimes label the opposite and adjacent sides incorrectly. Always draw the triangle and mark the angle θ first. If the side is not touching θ, it is opposite; if it touches θ and is not the hypotenuse, it is adjacent.
在三角学中，学生有时会错误地标注对边和邻边。一定要先画出三角形并标出角 θ。如果边不接触 θ，它就是对边；如果接触 θ 且不是斜边，它就是邻边。
Another common mistake is using degrees and radians incorrectly. For IGCSE, all trigonometric problems are in degrees unless stated otherwise. Make sure your calculator is set to degree mode.
另一个常见错误是错误地使用度数和弧度。对于 IGCSE，除非另有说明，所有三角学问题均以度为单位。请确保你的计算器设置为度数模式。
Finally, show all stages of your working. Circle-theorem and trigonometry questions often have several steps, and even if the final answer is wrong, method marks are available for correct formulas and substitutions.
最后，展示所有解题步骤。圆定理和三角学问题通常包含多个步骤，即使最终答案错误，正确的公式和代入步骤也可获得方法分。
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